Which Maxwell equation in differential form expresses Faraday's law of induction?

• A. ∇×E = −∂B/∂t
• B. ∇·E = ρ/ε0
• C. ∇×B = μ0 J + μ0 ε0 ∂E/∂t
• D. ∇·B = 0

Answer: (A)

A changing magnetic field generates a circulating electric field. In differential form, this is written as ∇×E = −∂B/∂t. The curl of E tells you how the electric field lines swirl around a point; a nonzero curl means there’s a local rotation of E. The time rate of change of B sets how strong that circulation is at each point. The negative sign embodies Lenz’s law: the induced electric field tends to oppose the change in magnetic flux through any loop, so the induced circulation opposes the change in B. In integral form, this becomes ∮ E · dl = −dΦ_B/dt.

The other equations are related but describe different aspects: one is Gauss’s law for electricity (divergence of E relates to charge density), another is Ampère–Maxwell’s law (curl of B relates to currents and changing E), and the last is Gauss’s law for magnetism (divergence of B is zero, meaning no magnetic monopoles).